Sequential Communication Bounds for Fast Linear Algebra

Abstract

In this note we obtain communication cost lower and upper bounds on the algorithms for LU and QR given in (Demmel, Dumitriu, and Holtz 2007). The algorithms there use fast, stable matrix multiplication as a subroutine and are shown to be as stable and as computationally efficient as the matrix multiplication subroutine. We show here that they are also as communication-efficient (in the sequential, two-level memory model) as the matrix multiplication algorithm. The analysis for LU and QR extends to all the algorithms in (Demmel, Dumitriu, and Holtz 2007). Further, we prove that in the case of using Strassen-like matrix multiplication, these algorithms are communication optimal. ∗This work is supported by Microsoft (Award #024263) and Intel (Award #024894) funding and by matching funding by U.C. Discovery (Award #DIG07-10227); additional support from Par Lab affiliates National Instruments, NEC, Nokia, NVIDIA, and Samsung. Research supported by U.S. Department of Energy grants under Grant Numbers DE-SC0003959, DE-SC0004938, and DE-FC02-06-ER25786, as well as Lawrence Berkeley National Laboratory Contract DE-AC02-05CH11231. Research supported by the Sofja Kovalevskaja programme of Alexander von Humboldt Foundation and by the National Science Foundation under agreement DMS-0635607. Research supported by ERC Starting Grant Number 239985.

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Cite this paper

@inproceedings{Ballard2012SequentialCB, title={Sequential Communication Bounds for Fast Linear Algebra}, author={Grey Ballard and James Demmel and Olga Holtz and Oded Schwartz}, year={2012} }